Question

Transmission tower antenna is at a height of 20m. How much range can it cover if the receiving antenna is at a height of 25m?

Answer 1

Hint: An antenna is basically an array of conductors which is electrically connected to the receiver or transmitter. With the help of the height of the transmission antenna and the height of the receiving antenna and the radius of the earth, the distance between the transmitter and receiver is first calculated. From this distance, the total range can be calculated.
Formula Used:
Distance between the transmitter antenna and the receiver antenna
=$\sqrt{2\times H(t)\times R}+\sqrt{2\times H(r)\times R}$

Complete answer:
Antennas are used to transmit and receive radio signals equally in all directions. In this problem, the height of both the transmission and the receiving antennas are given.
Given,
Height of the transmission antenna, H (t) =20 m
Height of the receiving antenna, H(r) =25m
We know, the radius of the earth, R = 6400 km
                                                            $\Rightarrow R=6.4\times {{10}^{6}}m$
To find: The Range that the antenna can cover
Therefore, the distance between the receiver antenna and the transmission antenna is given by
$\begin{align}
  & =\sqrt{2\times 6.4\times {{10}^{6}}\times 20}+\sqrt{2\times 6.4\times {{10}^{6}}\times 25} \\
 & =33888.54m \\
 & =33.888km \\
\end{align}$
The above gives us the distance between the transmission antenna and the receiving antenna. This is also known as the line of sight. From the line of sight, the total range can be calculated by calculating the area.
Range= Total area covered
$\Rightarrow $ R=$\pi \times {{(\text{Line of sight)}}^{2}}$
$\Rightarrow R=\pi {{(33.888)}^{2}}$
$\Rightarrow R=3609.246k{{m}^{2}}$
Hence, the total range covered by the antenna is given by 3609.246$k{{m}^{2}}$ .

Additional Information:
Antennas are mostly used in radio engineering to propagate radio waves through space and electric currents moving in metal conductors. The first antennas were built by Hertz to prove the existence of waves predicted by electromagnetic theory of James Clerk Maxwell. Marconi started using and developing antennas for long distance communication. He received the Nobel Prize for the same.

Note:
In this problem, we first calculate the line of sight of the antenna using three particular values. The 3 values are the radius of the earth, the height of the transmission antenna and the height of the receiving antenna. With the help of line of sight, we calculate the area covered by the antenna taking the line of sight as the radius. This area gives us the value of the range covered by the antenna. Care must be taken while calculating the line of sight, the formula might be mistaken.